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The problems and solutions from examination i of the es205 course, focusing on differential equations and system modeling. Students are required to determine the value of constants, find steady state values, estimate natural frequencies, and derive transfer functions. The problems involve second order systems, spring-mass systems, and electric circuits.
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Name 25 pts ES205 Examination I Problem 1 March 30, 2006
You must show all work for full credit on these problems.
1.1 (8 pts)The differential equation for a system is given by 100 x && + cx &+ 4 x = 20 y.
a) Determine the value of c so that the damping ratio is 1. b) The steady state value of x(t) if y is an input of amplitude 3.
1.2 (4 pts) A laboratory notebook for a secret project was burned in a fire. The only scrap of paper remaining is shown below with the handwritten comment from one of the investigators. From other documentation you know that this scrap refers a second order system. Determine the natural frequency of the system.
We know ζ = 0.3 so ω n =
Time (s)
Name 35 pts ES205 Examination I Problem 2 March 30, 2006
During your summer internship at NASA Glenn, you are tasked with finding a transfer function model of the Stirling rectilinear generator depicted below. You came up with the simplified schematic where the Stirling power piston provides a known position input to the Mover. The Mover is modeled as a rod with stiffness k 1 , connected to magnets of mass m. The mover is anchored to the ground by stiffness k 2 and damping b. The rectilinear alternator provides a current source with i = Kx &. Signal conditioning is provided by the cascaded op-amps. Find the equations needed to derive a transfer function from input Y ( s ) to output E 2 ( s ). Your solution should consist of a clear set of equation and a list of unknowns.
y ( t )
m
k 1
k 2
b
x ( t ) R 1 R 1
e^ C^1 e 2 i = Kx &^ e (^01)
R 2
R 1
Name 40 pts ES205 Examination I Problem 3 March 30, 2006
The field and armature windings of an electric motor are connected in parallel across a voltage source eF (t), as shown in the figure below. The resistances of the field and armature windings are Rf and RA respectively. The inductance of the field and armature windings are LF and LA respectively. a) Determine the equations that govern this system. Clearly document your solution and keep track of equations and unknowns. b) Write these equations as a set of first order differential equations (6 pts).
RA
RF L (^) A
L (^) F
eF
+
-
I (^) A
J (^) M k
J (^) L I (^) F
c
